Here w⁡(x) is continuous or piecewise continuous or integrable, and such that
0<∫abx2⁢n⁢w⁡(x)⁢dx<∞ for all n.

It is assumed throughout this chapter that for each polynomial pn⁡(x)
that is orthogonal on an open interval (a,b) the variable x is confined to
the closure of (a,b)unless indicated otherwise.
(However, under appropriate conditions almost all equations given in the chapter
can be continued analytically to various complex values of the variables.)

Orthogonality on Finite Point Sets

Let X be a finite set of distinct points on ℝ, or a countable infinite
set of distinct points on ℝ, and wx, x∈X, be a set of positive
constants. Then a system of polynomials {pn⁡(x)}, n=0,1,2,…, is
said to be orthogonal on X with respect to the weightswx if

More generally than (18.2.1)–(18.2.3),
w⁡(x)⁢dx may be replaced in (18.2.1) by a positive measure
dα⁡(x), where α⁡(x) is a bounded nondecreasing function on
the closure of (a,b) with an infinite number of points of increase, and such
that 0<∫abx2⁢n⁢dα⁡(x)<∞ for all n. See
McDonald and Weiss (1999, Chapters 3, 4) and Szegő (1975, §1.4).

§18.2(ii) x-Difference Operators

If the orthogonality discrete set X is {0,1,…,N} or
{0,1,2,…}, then the role of the differentiation operator d/dx
in the case of classical OP’s (§18.3) is played by Δx,
the forward-difference operator, or by ∇x, the backward-difference
operator; compare §18.1(i). This happens, for example, with the
Hahn class OP’s (§18.20(i)).

If the orthogonality interval is (-∞,∞) or (0,∞), then the
role of d/dx can be played by δx, the central-difference
operator in the imaginary direction (§18.1(i)). This happens, for
example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials
(§18.20(i)).

If the OP’s are orthonormal, then cn=an-1 (n≥1).
If the OP’s are monic, then an=1 (n≥0).

Conversely, if a system of polynomials {pn⁡(x)} satisfies
(18.2.10) with an-1⁢cn>0 (n≥1), then {pn⁡(x)}
is orthogonal with respect to some positive measure on ℝ (Favard’s
theorem). The measure is not necessarily of the form w⁡(x)⁢dx nor is it
necessarily unique.

All n zeros of an OP pn⁡(x) are simple, and they are located in the
interval of orthogonality (a,b).
The zeros of pn⁡(x) and pn+1⁡(x) separate each other, and if m<n then
between any two zeros of pm⁡(x) there is at least one zero of pn⁡(x).